{"status": "success", "data": {"description_md": "In a small pond there are eleven lily pads in a row labeled $0$ through $10$. A frog is sitting on pad $1$. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\\frac{N}{10}$ and to pad $N+1$ with probability $1-\\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad $0$ it will be eaten by a patiently waiting snake. If the frog reaches pad $10$ it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake?\n\n$\\textbf {(A) } \\frac{32}{79} \\qquad \\textbf {(B) } \\frac{161}{384} \\qquad \\textbf {(C) } \\frac{63}{146} \\qquad \\textbf {(D) } \\frac{7}{16} \\qquad \\textbf {(E) } \\frac{1}{2}$", "description_html": "<p>In a small pond there are eleven lily pads in a row labeled  <span class=\"katex--inline\">0</span>  through  <span class=\"katex--inline\">10</span> . A frog is sitting on pad  <span class=\"katex--inline\">1</span> . When the frog is on pad  <span class=\"katex--inline\">N</span> ,  <span class=\"katex--inline\">0&lt;N&lt;10</span> , it will jump to pad  <span class=\"katex--inline\">N-1</span>  with probability  <span class=\"katex--inline\">\\frac{N}{10}</span>  and to pad  <span class=\"katex--inline\">N+1</span>  with probability  <span class=\"katex--inline\">1-\\frac{N}{10}</span> . Each jump is independent of the previous jumps. If the frog reaches pad  <span class=\"katex--inline\">0</span>  it will be eaten by a patiently waiting snake. If the frog reaches pad  <span class=\"katex--inline\">10</span>  it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake?</p>\n<p> <span class=\"katex--inline\">\\textbf {(A) } \\frac{32}{79} \\qquad \\textbf {(B) } \\frac{161}{384} \\qquad \\textbf {(C) } \\frac{63}{146} \\qquad \\textbf {(D) } \\frac{7}{16} \\qquad \\textbf {(E) } \\frac{1}{2}</span> </p>\n<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "2014 AMC 10B Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/14_amc10B_p24"}}